Research projects for 2008/2009

 

Note: More students can choose the same topic, BUT they should work independently and should have independent results.

 

 

1.       Stochastic resonance in a 1D spring-block chain.

 

The phenomenon of stochastic-resonance [1] will be studied in the original 1D spring-block model introduced in [2]. Beside the forces considered in [1] an additional random force is considered and the behaviour of the system in oscillating magnetic field is studied as a function of the strength of the random forces.

[1] Stochastic resonance. (stoch-res0.pdf , stoch-res.pdf, stoch-res2.pdf)

[2]  Kovács, K., Brechet, Y., Néda, Z. (2005): A spring-block model for Barkhausen noise — Modelling and Simulation in Mat. Sci. Eng. 13(8) pp 1341-1352 (spring-block0.pdf , spring-block1.pdf)

 

2.     Stochastic spring-block model for glass-breaking phenomena. The case of a glass-plate with uniformly applied shock.

 

Spring-block models will be used to model breaking of glass-plates under uniform shock-loading. From modelling we are interested of the resulting fragmentation pattern and fragment-size distribution. Simulation results will be compared with experimental ones. (spring-block0.pdf , spring-block2.pdf) 

 

3.     Pulling out entangled fibres – modelling with simple models.

 

Entangled materials are usually composed of long-fibres…as an example see cotton or wool. When pulling out a fibre from it the force will vary in time. This project will simulate the force-varying pattern and will compare the results with experimental ones. Either a simple spring-block model [1] or a more complicated collective behaviour model [2] will be considered.

 

[1] Spring-block models(spring-block0.pdf)

[2] a more complicated model (entangled.pdf)

 

4.     Spring-block models on random graphs.  

 

We will construct a random graph connected by springs and blocks. The vertices will be blocks and the links between them springs. The system can move (pulled by the first vertex) on a chess-board like pattern, where the friction varies from cell to cell. We are interested on the motion of this graph-like spring-block system in several aspects. How does the configuration of the chain changes in time (graphics) and how does the largest size of the cluster fluctuate in time? How does the tension in the springs fluctuate in time? (The equilibrium distance of the springs is not zero)

 

[1] spring-block models (spring-block0.pdf)

5.     Random asset-exchange models for understanding the universal Pareto-Zipf’s law.

 

The Pareto-Zipf distribution [1] is one of the most general distribution in social and economic systems. Shortly it states that the rank-abundance curve for various quantities follows a power-law tail. The aim of this research project is to use random-asset exchange models [2] to simulate it’s origin. Monte Carlo type simulations will be performed and the results compared to experimental data on wealth distribution and settlement-size distribution.

 

[1] W.J. Reed; The Pareto, Zipf and other power laws (pareto.pdf)

[2]  A. Chatterjee, S. Sinha and B.K. Chakrabarti; Economic inequality: Is it natural? (asset-exchange.pdf , sinha1.pdf)

 

6.     Magnetic properties of Ising models on random graphs.

 

The aim of this research project is to investigate the magnetic behaviour of Ising spins placed on random graphs. We will use the classical Metropolis algorithm and the cluster algorithms (Swensen and Wang, Wolf ….) to study the magnetic properties of such systems [1]. We are interested in special in the case of scale-free networks [2].

 

[1] see the course material ….Z. Neda…

[2] for networks see the www.nd.edu/~networks homepage or the review paper by A.L. Barabasi and R. Albert (scalefree.pdf)   

 

7.      Two-mode stochastic oscillators on random graphs.

 

Stochastic oscillators with several possible co-existing modes present an exciting system with interesting non-trivial collective behaviour [1]. Synchronization can also appear, and the periodicity of the whole system is greatly enhanced in the case of global coupling. The aim of this research project will be to study the case when the interactions are on a random-graph-like topology. We are especially interested in the case of scale-free networks [2].

 

[1] papers on multi-mode stochastic oscillators (twomode.pdf , multimode.pdf)

[2] for networks see the www.nd.edu/~networks homepage or the review paper by A.L. Barabasi and R. Albert (scalefree.pdf)   

 

8.     The graph colouring problem on scale-free networks

 

The graph colouring problem [1] is a well-known NP hard problem, which means that the time necessary for solution scales faster than a polynomial with system size. The aim of this research project is to study (make an algorithm) the  graph-colouring on scale-free networks [2].

 

[1] Algorithms for graph colouring (graphcolouring.pdf)

[2] for networks see the www.nd.edu/~networks homepage or the review paper by A.L. Barabasi and R. Albert (scalefree.pdf)    

9.     Correlations in the stock-market. Analysis of random fluctuations.

 

Within this research project one will analyze fluctuation in stock indexes and correlations in the index components. For correlations we will use the correlation matrix, and study correlation investigating the eigen-value spectra of this. The DJ, BUX and BET stock-indexes will be in the focus of our study [1].

 

[1] The inverse statistics method and assumptions on correlation between stock components (stock1.pdf , stock2.pdf)  

 

10.  Statistical properties of growing Voronoi cells in 1D and 2D.

 

The size-distribution of Poissonian Voronoi cells were investigated by many previous studies [1]. In the present study we will focus on the statistics of Voronoi type cells, which are obtained as a result of a growth process from randomly distributed nucleation centres. In case when all random nucleation points are generated in the same time-moment one will naturally regain the classical Poissonian cells. We are interested how this distribution will change when the nucleation centres are constantly appearing in time…..

 

 [1] study on Poissonian Voronoi cells size distribution (voronoi.pdf)

 

11.  Statistical and computer simulation study on the minority game problem.

     

      The problem that it is proposed here [1] is a classical problem from game-theory. It is interesting from several aspects for the physics community: self-organization, connection with NP hard statistical physics optimizations and critical phenomena. It started from the classical El-Farol bar problem [2]. El-Farol is an Irish  bar in Santa Fe where excellent music is played. The problem is that it is many times over-crowded and if there are too many peoples the atmosphere isn’t great. So everybody has to make a decision based on the past history of how the bar was visited. The decision is to go or not….if they go and it is overcrowded it is a wrong decision since the atmosphere will be bad, and again if they do not go and there are not many peoples there it was also a wrong decision. The question is which is the best strategy (if there is one) based on the past history to get the optimal decision.

 

[1] the history of minority game problems (minority-game1.pdf)

[2] The El-Farol problem (minority-game2.pdf)

 

12.  Simulating the secretary problem.

 

    The secretary problem is an optimal stopping problem that has been studied extensively in the fields of applied probability, statistics, and decision theory. It is also known as the marriage problem, the sultan's dowry problem, the fussy suitor problem, and the best choice problem. The problem can be stated as follows: (i) There is a single secretarial position to fill. (2) There are n applicants for the position, and the value of n is known. (3) The applicants can be ranked from best to worst with no ties. (4) The applicants are interviewed sequentially in a random order, with each order being equally likely.(5) After each interview, the applicant is accepted or rejected. (6) The decision to accept or reject an applicant can be based only on the relative ranks of the applicants interviewed so far. (7) Rejected applicants cannot be recalled. (8) The object is to select the best applicant.

 

One reason why the secretary problem has received so much attention is that the optimal policy for the problem (the stopping rule) has a surprising feature. Specifically, for large n the optimal policy is to skip the first n / e applicants (where e is the base of the natural logarithm) and then to accept the next candidate---an applicant that is better than all those previously interviewed. As n gets larger, the probability of selecting the best applicant from the pool goes to 1 / e, which is around 37%. Whether one is searching through 100 or 100,000,000 applicants, the optimal policy will select the single best one about 37% of the time.

 

The problem has an analytical solution as well [1]. The aim of this research project is to simulate the problem and show that the above described solution is the optimal one!

 

[1] a review article on the secretary problem (secretary.pdf)

 

13.  Stochastic simulations of social systems with STARLOGO

 

StarLogo [1] is a user-friendly and easy-to use simulation software. It is a tool to create and understand simulations of complex systems, it also brings with it several advances: (1) lower the barrier to entry for programming with a graphical interface where language elements are represented by coloured blocks that fit together like puzzle pieces, (2) entice more young people into programming through tools that facilitate making games, (3) use 3D graphics to make more compelling and rich games and simulation models. The aim of this research project is to use StarLogo for simulating several simple interacting social or biological systems where non-trivial collective behaviour appears: flocking, synchronization, phase-transitions and pattern formation.

 

 [1] http://education.mit.edu/drupal/starlogo-tng